Pascal, Pacioli, Probability, and Problem-Based Learning

I’m currently reading Howard Eves’s Great Moments in Mathematics After 1650 (1983, Mathematical Association of America), a chronological collection of lectures. The first lecture in this volume (the second of two) is on the development of probability as a formal field of mathematics as it was driven by Pascal and Fermat, with regards to a specific problem put in print by Pacioli:

Two people are playing a game. For each round, each player has equal chance of getting a point. The first player wins if he’s ahead by two points; the second player wins if he’s ahead by three points. What is the likelihood that the first player will win?

Using modern mathematical techniques, this is a moderate to difficult application of probability. However, according to Eves, Pacioli and several other answers were incorrect, and it wasn’t until Pascal and Fermat set themselves to it that a definitive answer was derived (with two explanations).

Pascal, for his part, used his namesake triangle. The fifth line represented the relative probabilities: 1 4 6 4 1, giving the first player 11 wins to the second player’s 5. This approach allowed Pascal to general his solution to any set of points. If the first player has to win by 3 while the second has to win by 5, use the eighth row.

As a teacher, one of my responses was to notice the gap of time between Pacioli’s presentation and the solution. Pacioli published in 1494, while Pascal and Fermat agreed on a solution in 1654: 170 years later. Nowadays, the problem might take a session or two in a probability course; originally, it took nearly two centuries.

At the extreme end of inquiry-based learning, this is a sobering reality. Pascal and Fermat were undeniably geniuses at mathematics, and many others had in that time period addressed the problem and failed. To the extent that we set students loose with problems and encourage them to solve them independently, we’re looking to wait 170 years for solutions.

Naturally, inquiry-based, problem-based, and project-based exercises should come with appropriate scaffolding, and examples like this from the history of mathematics (this is but one of many) illustrate the importance of that scaffolding. In our enthusiasm to move away from lecture-based instruction, it’s crucial to keep in mind that what seems straightforward to us, the content experts, took a very long time to develop, even by people dedicated to the task.